Analysis III

Question 1

Define a step function

Answer 1

If [a, b] isan interval, then a function φ: [a, b] -> R is a step function if there is a finite sequence a = x0 ≤ x1 ≤…≤ xn = b such that φ is constant on each open interval (x(i−1), xi).

Question 2

Define I(φ)

Answer 2

If φ is a step function adapted to some partition P, where φ(x) = ci on the interval (xi−1, xi), then I(φ) = the sum of ci(xi-x(i-1)).

Question 3

Define a minorant

Answer 3

A step function φ− is a minorant for f if f ≥ φ− pointwise.

Question 4

Define a majorant

Answer 4

A step function φ+ is a majorant for f if f ≤ φ+ pointwise.